Abstract
We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k ≥ 4. For k ≥ 6 our constructions have \(> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2\) edges, which is asymptotically best possible by Turán’s theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length ≤ ℓ for any ℓ and any k≥4, again with a best possible density of \(> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2\) edges for k ≥ 6. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the correct maximal density of k-critical members (k ≥ 6).
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Pegden, W. Critical graphs without triangles: An optimum density construction. Combinatorica 33, 495–512 (2013). https://doi.org/10.1007/s00493-013-2440-1
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DOI: https://doi.org/10.1007/s00493-013-2440-1